\section{Q3: Two-Factor Interest Rate Models}
Let 
\begin{align}
r(t) = X_1(t) + X_2(t),
\end{align}
where $X_1(t)$ and $X_2(t)$ are Ornstein-Uhlenbeck processes;
\begin{align}
dX_i(t) = \kappa_i \paren{\theta_i - X_i(t)}dt + \sigma_i\dwq_i,
\end{align}
where $W_i$ are $\mQ$-Brownian motions.
\subsection{Q.3a}
We derive a formula for ZCB prices in the model given above. Consider
\begin{align}
\ptt = E^\mQ\brackk{e^{-\int_t^T r(s) ds}} = E^\mQ_t\brackk{e^{-\int_t^T X_1(s) ds}}E^\mQ_t\brackk{e^{-\int_t^T X_2(s) ds}} = P_1(t,T)P_2(t,T),
\end{align}
since $X_1$ and $X_2$ are independent.
Thus we may calculate the price of the ZCB as a product of prices of the ZCBs $P_1$ and $P_2$. It follows from Bj\"ork proposition 24.3 that 
\begin{align}
P_i&(t,T) = \exp\paren{A_i(t,T) - B_i(t,T)X_i(t)}\nonumber\\ 
& =\exp{\paren{\frac{\frac{1}{\kappa_i}\paren{1-e^{-\kappa_i(T-t)}-T+t}\paren{\theta_i\kappa_i^2-\half\sigma_i^2}}{\kappa_i^2} - \frac{\sigma_i^2\frac{1}{\kappa_i^2}\paren{1-e^{-\kappa_i(T-t)}}^2}{4\kappa_i}X_i(t)}},
\end{align}
where 
\begin{align}
B_i(t,T) & = \frac{1}{\kappa_i}\paren{1-e^{-\kappa_i(T-t)}},\nonumber\\
A_i(t,T) & = \frac{\paren{B_i(t,T) - T + t}\paren{\kappa_i^2\theta_i - \half \sigma_i^2}}{\kappa_i^2} - \frac{\sigma_i^2B_i^2(t,T)}{4\kappa_i}.\nonumber
\end{align}
Then
\begin{align}
P&(t,T) = \exp\paren{\sum_{i=1}^2{A_i(t,T) - B_i(t,T)X_i(t)}}\nonumber\\ 
& =\exp{\paren{\sum_{i=1}^2\frac{\paren{\frac{1}{\kappa_i}(1-e^{-\kappa_i(T-t)})-T+t}\paren{\theta_i\kappa_i^2-\half\sigma_i^2}}{\kappa_i^2} - \frac{\sigma_i^2\frac{1}{\kappa_i^2}\paren{1-e^{-\kappa_i(T-t)}}^2}{4\kappa_i}X_i(t)}}.
\end{align}
\subsection{Q.3b}
We calculate correlations between changes over short time horizons in ZCB yields. Now
\begin{align}
y(t,T) & = -\frac{\log P(t,T)}{T-t} = -\frac{\log P_1(t,T) + \log P_2(t,T)}{T-t}\nonumber\\
& = -\frac{A_1(t,T) + A_2(t,T) - B_1(t,T)X_1(t) - B_2(t,T)X_2(t)}{T-t}\nonumber\\
& = -\frac{g(t, X_1(t), X_2(t),T)}{T-t},
\end{align}
where $g(t, X_1(t), X_2(t),T) = A_1(t,T) + A_2(t,T) - B_1(t,T)X_1(t) - B_2(t,T)X_2(t)$.
Let $\tau_1$ and $\tau_2$ be two different times of maturity and consider $\Delta y(t,\tau_i) = y(t+\Delta t,\tau_i) - y(t,\tau_i)\simeq dy(t,\tau_i)$. Then by It\=o we have
\begin{align}
dy(t,\tau_i) & = \paren{\frac{1}{(\tau_i-t)^2}g(t,X_1(t),X_2(t),\tau_i) - \frac{1}{\tau_i-t}g_t(t,X_1(t),X_2(t),\tau_i) - \sum_{j=1}^2\frac{B_j(t,\tau_i)}{\tau_i-t}\kappa_j\paren{\theta_j - X_j(t)}}dt \nonumber\\
&- \sigma_1\frac{B_1(t,\tau_i)}{\tau_i-t}\dwq_1 - \sigma_2\frac{B_2(t,\tau_i)}{\tau_i-t}\dwq_2\nonumber\\
& = h(t,X_1(t),X_2(t),\tau_i)dt - \sigma_1\frac{B_1(t,\tau_i)}{\tau_i-t}\dwq_1 - \sigma_2\frac{B_2(t,\tau_i)}{\tau_i-t}\dwq_2
\end{align}
with $g_t = \pdiff{g}{t}$. Then, ignoring terms of order $O(dt^2)$ and $O(\dw dt)$ we have
\begin{align}
dy(t,\tau_1)dy(t,\tau_2) & = \sigma^2_1\frac{B_1(t,\tau_1)B_1(t,\tau_2)}{(\tau_1-t)(\tau_2-t)}dt + \sigma^2_2\frac{B_2(t,\tau_1)B_2(t,\tau_2)}{(\tau_1-t)(\tau_2-t)}dt.\nonumber
\end{align}
Furthermore we have
\begin{align}
E^\mQ_t\brackk{dy(t,\tau_i)} = h(t,X_1(t),X_2(t),\tau_i)dt\label{eq:dy:expect}
\end{align}
since $E^\mQ_t\brackk{\dwq_i} = 0$. Thus
\begin{align}
E^\mQ_t\brackk{dy(t,\tau_1)}E^\mQ_t\brackk{dy(t,\tau_2)} = O(dt^2).
\end{align}
Then, again ignoring terms of order $O(dt^2)$ and $O(\dw dt)$, we have
\begin{align}
Cov\paren{dy(t,\tau_1),dy(t,\tau_2)} & = E^\mQ_t\brackk{dy(t,\tau_1)dy(t,\tau_2)} - E^\mQ_t\brackk{dy(t,\tau_1)}E^\mQ_t\brackk{dy(t,\tau_2)}\nonumber\\
& \simeq  \sigma^2_1\frac{B_1(t,\tau_1)B_1(t,\tau_2)}{(\tau_1-t)(\tau_2-t)}dt + \sigma^2_2\frac{B_2(t,\tau_1)B_2(t,\tau_2)}{(\tau_1-t)(\tau_2-t)}dt.
\end{align}
Furthermore we have
\begin{align}
Var\paren{dy(t,\tau_i)} & = E^\mQ\brackk{dy(t,\tau_i)^2} - \paren{E^\mQ\brackk{dy(t,\tau_i)}}^2 \nonumber\\
& \simeq \sigma^2_1\frac{B^2_1(t,\tau_i)}{(\tau_i-t)^2}dt + \sigma^2_2\frac{B^2_2(t,\tau_i)}{(\tau_i-t)^2}dt,
\end{align}
where as earlier we have neglected terms of order $O(dt^2)$.
Finally we have
\begin{align}
Corr\paren{dy(t,\tau_1	), dy(t,\tau_2)} & = \frac{Cov\paren{dy(t,\tau_1),dy(t,\tau_2)}}{\sqrt{Var\paren{dy(t,\tau_2)}Var\paren{dy(t,\tau_2)}}}\nonumber\\
& = \frac{\sigma^2_1\frac{B_1(t,\tau_1)B_1(t,\tau_2)}{(\tau_1-t)(\tau_2-t)} + \sigma^2_2\frac{B_2(t,\tau_1)B_2(t,\tau_2)}{(\tau_1-t)(\tau_2-t)}}{\sqrt{\sigma^2_1\frac{B^2_1(t,\tau_1)}{(\tau_1-t)^2} + \sigma^2_2\frac{B^2_2(t,\tau_1)}{(\tau_1-t)^2}}\sqrt{\sigma^2_1\frac{B^2_1(t,\tau_2)}{(\tau_2-t)^2} + \sigma^2_2\frac{B^2_2(t,\tau_2)}{(\tau_2-t)^2}}}
\end{align}
which was what we desired to calculate.
\subsection{Q.3c}
As Bj\"ork notes on page 410 (top of page) of `Arbitrage Theory in Continuous Time' proposition 26.13 does indeed hold for the Vasi\v cek model, since this model is a special case of the Hull-White model.
Since we are in the two-factor case, let us consider the following
\begin{align}
Z(t) = \frac{P(t,\tau_2)}{P(t,\tau_1)} = \frac{P_1(t,\tau_2)P_2(t,\tau_2)}{P_1(t,\tau_1)P_2(t,\tau_1)} \label{eq:z_vasicek}
\end{align}
where we used the notation of Q.3a. Now with $Z_i \paren{t}= P_i(t,\tau_2)/P_i(t,\tau_1)$, we see that
\begin{align}
dZ(t) & = Z_1(t)dZ_2(t) + Z_2(t)dZ_1(t) + dZ_1dZ_2\nonumber\\
& = \zt\zto\paren{\ldots}dt + \zt\zto\paren{\sigma_{Z_1}(t)\dwq_1 + \sigma_{Z_2}(t)\dwq_2}\nonumber\\ 
& + \paren{\zt dt + \zt\sigma_{Z_1}(t)\dwq_1}\paren{\zto dt + \zt\sigma_{Z_2}(t)\dwq_2}\nonumber\\
& = \zt\zto\paren{\ldots}dt + \zt\zto\paren{\sigma_{Z_1}(t)\dwq_1 + \sigma_{Z_2}(t)\dwq_2}\nonumber\\
& = Z(t)\paren{\ldots}dt + Z(t)\sigma^*_Z(t)\dw
\end{align}
since $dZ_i(t) = Z_i(t)\paren{\ldots}dt + Z_i(t)\sigma_{Z_i}(t)\dwq_i$ (see Bj\"ork p. 409, equation 26.44). In the above we used the following
\begin{align}
& \sigma_Z(t)  = \binom{\sigma_{Z_1}(t)}{\sigma_{Z_2}(t)},\nonumber\\
& \dw 						   = \binom{\dwq_1}{\dwq_2},\nonumber
\end{align}
with 
\begin{align}
\sigma_{Z_i}(t) = \frac{\sigma_i}{\kappa_i}e^{\kappa_i t}\paren{e^{-\kappa_i \tau_2} - e^{\kappa_i \tau_1}}.
\end{align}
Thus the two-factor Vasi\v cek model satisfies assumption 26.5.1 of Bj\"ork (p. 407). Let
\begin{align}
\Sigma^2(t,\tau_1) & = \int_{t}^{\tau_1} \lVert\sigma_Z\paren{s}\rVert^2 ds\nonumber\\
& = \frac{\sigma_1^2}{2\kappa_1^3}\paren{1-e^{-2\kappa_1\paren{\tau_1-t}}}\paren{1-e^{-\kappa_1\paren{\tau_2 - \tau_1}}}^2 + \frac{\sigma_2^2}{2\kappa_2^3}\paren{1-e^{-2\kappa_2\paren{\tau_1-t}}}\paren{1-e^{-\kappa_2\paren{\tau_2 - \tau_1}}}^2.\nonumber
\end{align}
Then, again following the logic of proposition 26.13, we have the following price for a call-option, with expiry $\tau_1$, on a ZCB with maturity $\tau_2$ and strike $K$ is
\begin{align}
\Pi\paren{t;P(t,\tau_2)} = P_1(t,\tau_2)P_2(t,\tau_2)\Phi\paren{d_1} - P_1(t,\tau_1)P_2(t,\tau_1) K \Phi\paren{d_2}
\end{align}
where 
\begin{align}
d_2 & = \frac{\log\paren{\frac{P(t,\tau_2)}{KP(t,\tau_1)}} - \half\Sigma^2(t,\tau_1)}{\sqrt{\Sigma^2(t,\tau_1)}}\nonumber\\
d_1 & = d_1 + \sqrt{\Sigma^2(t,\tau_1)}.
\end{align}

\subsection{Q.3d}
We show that $dr(t) = \tilde{\kappa}\paren{\tilde{\theta} - r(t)}dt + \tilde{\sigma}d\tilde{W}$, for a suitable choise of $\tilde{\kappa}$ and $\tilde{\theta}$, and where $\tilde{\theta}$ is in fact an Ornstein-Uhlenbeck proces itself. Using the hint so generously provided we write
\begin{align}
dr(t) = dX_1(t) + dX_2(t) & = \kappa_1\paren{\theta_1-X_1(t)}dt + \sigma_1\dwq_1 + \kappa_2\paren{\theta_2-X_2(t)}dt + \sigma_2\dwq_2 \nonumber\\
& = \kappa_1\paren{\theta_1-\paren{r(t)-X_2(t)}}dt + \sigma_1\dwq_1 + \kappa_2\paren{\theta_2-X_2(t)}dt + \sigma_2\dwq_2 \nonumber\\
& = \kappa_1\paren{\theta_1+\frac{\kappa_2}{\kappa_1}\theta_2 + \paren{1-\frac{\kappa_2}{\kappa_1}}X_2(t)-r(t)}dt + \sigma_1\dwq_1 + \sigma_2\dwq_2 \nonumber\\
& = \tilde{\kappa}\paren{\tilde{\theta}-r(t)}dt + \tilde\sigma d\tilde{W}, \nonumber\\
\end{align}
where
\begin{align}
\tilde{\kappa} & = \kappa_1,\nonumber\\
\tilde{\theta} & = \theta_1+\frac{\kappa_2}{\kappa_1}\theta_2 + \paren{1-\frac{\kappa_2}{\kappa_1}}X_2(t),\nonumber\\
\tilde{W} & = \frac{1}{\sqrt{\sigma_1^2 + \sigma_2^2}}\paren{\sigma_1\wq_1 + \sigma_2\wq_2},\nonumber\\
\tilde{\sigma} & = \sqrt{\sigma_1^2 + \sigma_2^2}.\nonumber
\end{align}
We see that $\tilde{\theta}$ is itself an Ornstein-Uhlenbeck process. Indeed
\begin{align}
d\tilde{\theta} & = \paren{1-\frac{\kappa_2}{\kappa_1}}dX_2(t)\nonumber\\
& = \kappa_2\paren{1-\frac{\kappa_2}{\kappa_1}}\paren{\theta_2 - X_2(t)}dt + \paren{1-\frac{\kappa_2}{\kappa_1}}\sigma_2\dwq_2\nonumber\\
& = \kappa_2\paren{\theta_2 + \theta_1 - \paren{\theta_1 + \frac{\kappa_2}{\kappa_1}\theta_2 + \paren{1-\frac{\kappa_2}{\kappa_1}X_2(t)}}}dt + \sigma_{\tilde{\theta}}\dwq_2\nonumber\\
& = \kappa_2\paren{\theta_2 + \theta_1 - \tilde{\theta}}dt + \sigma_{\tilde{\theta}}\dwq_2,
\end{align}
where $\sigma_{\tilde{\theta}} = \paren{1-\frac{\kappa_2}{\kappa_1}}\sigma_2$. This was what we wanted to show.
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